Discussion
The value of \(\int_{-2}^{1} \frac{|x|}{x}dx\) is equal to
Answer: A
The value is -1.
\(\frac{|x|}{x}\) is -1 for x < 0 and +1 for x > 0, so split at 0.
\[ \begin{aligned} \int_{-2}^{1}\frac{|x|}{x}dx &= \int_{-2}^{0}(-1)\,dx + \int_{0}^{1}dx \\ &= -2 + 1 = -1 \end{aligned} \]
Treating \(\frac{|x|}{x}\) as 1 throughout gives 3, the length of the interval, and treating it as -1 throughout gives -3. Geometrically, the integrand is a step: the area below the axis on [-2, 0] is 2 and the area above it on [0, 1] is 1.
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